Genetics of Mutualism: the Evolution of Altruism Between Species

J. theor. Biol. (1994) 170, 393-400

Genetics of Mutualism: The Evolution of Altruism between Species

STEVEN A. FRANK

Department of Ecology and Evolutionary Biology, University of California, Irvine, CA 92717, U.S.A.

(Received on 20 January 1994, Accepted on 6 June 1994)

Conditions are analyzed under which natural selection favors an individual to help another species at a cost to its own reproduction. Traditional models for the evolution of altruism between species focus on the genetic relatedness between the original donor and the recipients of return benefits from the mutualistic partner species. A more general model is analyzed here that focuses on the synergistic effects between partner species caused by genetic variability. The model shows that the spread of altruism is enhanced by spatial correlations between species in the genetic tendency to give aid to partners. These spatial correlations between species are similar to the kin selection coefficients of relatedness that determine the course of social evolution within species. The model also shows that natural selection and ecological dynamics can create genetic correlations between neighbors of different species, even when the initial spatial distributions of the species are uncorrelated. Genetic correlations between species may play an important role in the origin and maintenance of altruism between species.

1. Introduction 1983b ), the process itself is unambiguous (Hamilton, 1975). Why would an individual sacrifice its own direct This paper presents models in which both species of reproduction to aid a member of another species? a mutualistic pair vary genetically in their tendency to Williams (1966) suggested that mutualistic aid of this aid their partners. Two interesting results emerge. sort is unlikely to evolve by natural selection. The First, the correlation between species in the tendency problem is that, even if the recipient returned benefits to aid partners plays an important role. The corre- to the donor species, the original donor individual lation between mutualistic partners has an effect would rarely receive those benefits. similar to the genetic correlations within species Hamilton (1972) and Wilson (1980, 1983a) agreed that form the standard coefficients of relatedness with Williams's logic but argued that, in many natural in kin selection models. Positive correlations systems, limited dispersal keeps individuals of both between species can greatly enhance the spread of species in close proximity. Thus aid given to another mutualism. species is likely to be returned to the original donor The second result concerns how correlations be- (Trivers, 1971) or its immediate neighbors. Because, tween species may arise. Correlations between neigh- under limited dispersal, the neighborhood constitutes bors of the same species are easily understood: a kin group, returned benefits enhance the inclusive neighbors are often kin, and kin share common fitness of the original donor. ancestors. By contrast, correlations between species The Hamilton–Wilson argument focused on gen- must be created by special migration or selection etic variability in the donor, with kin or group processes. In the models presented here, the synergis- selection as the mechanism by which mutualism tic effect of mutual aid between species creates posi- can evolve. Although there is some disagreement tive spatial correlations in the tendency to give aid to about whether "group selection" or "kin selection" mutualistic partners. Wilson (1993) has recently is the proper phrase for this scenario (Wilson, suggested that a similar process may occur at the

TABLE 1 The variance among groups in the average tendency of mutualists to give aid, the variance of js Definitions of notation used throughout this paper Symbol Definition Build-up of genetic correlation between species Regression coefficient of the total host aid provided to Genetic variability only in the host mutualists in group s on individual mutualist genotype in Genotypic value of a host individual. The amount of aid that group, j. Describes the association across space given by an individual increases linearly with its genotypic between mutualist genotype and total aid provided by value. hosts to mutualists. is Average genotypic value of hosts in the s-th subgroup of the population. Symbols are listed according to the section of the paper in which The average genotypic value over all hosts in the popu- they are first introduced. The symbol b, 's meaning changes slightly lation. in different sections and is therefore listed twice. Hs; The abundance of the i-th host genotype in the s-th group. h„, The absolute fitness of the i-th host genotype in the s-th group. community level, in which ecological dynamics create E The average fitness of hosts over all genotypes and groups. Relative fitness of the i-th genotype in the s-th group is a positive spatial correlation between mutualistic hsi I species. This community-level problem remains un- c The cost per unit aid given to a partner species. solved, however. b The fitness benefit per unit aid received from a partner species. To analyze the evolution of ecological interactions Regression coefficient of the average group values for the between species, the population genetic tools that hosts, i, on the individual values, i. This is the kin selection have been used to study social evolution are applied coefficient of relatedness among hosts within each group. here. This approach puts ecological analysis in an evolutionary context, providing mechanisms for the Genetic variability in both host and mutualist b, The fitness benefit to a host individual per unit of aid coevolution of ecological traits in a way that empha- available in the local group from other hosts. These are sizes social interactions. Simultaneously, the ecologi- indirect benefits received via the mutualist species, as shown in Fig. 1. cal side of social evolution is made clearer, because b2 The fitness benefit to a host individual per unit of aid genetic models traditionally analyze genotype fre- available in the local group from the mutualists. quency but ignore abundance. This expanded analysis is Average genotypic value of mutualists in the s-th group of the population. Genotypic value determines the amount of provides a more fluid notion of sociality. In the aid given to hosts. Discussion I consider a wider interpretation of co- Regression coefficient of the mutualists' average genotypic efficients of relatedness, and the question of whether value for groups s on individual host genotypes, i. De- scribes the genotypic association across space between the natural selection can favor the functional coherence hosts' and mutualists' tendency to give aid to each other. of interacting species.

The role of abundance Hs Total abundance of hosts in s -th group, H,=EiHsi. 2. Models Afsj The abundance of the j-th mutualist genotype in the s-th group. 2. 1 . GENETIC VARIABILITY ONLY IN THE HOST M, Total abundance of mutualists in the s-th group, Suppose that an individual of a host species pro-

4s Total aid provided by the hosts to the mutualists in the vides aid to a potentially mutualistic species. The s-th group, Os = Hsi,. fitness advantage for providing this aid depends on Ps Total aid provided by the mutualists to the hosts in the s-th group, AN = Msj„. the additional benefits that the mutualist returns to rd,51 Regression coefficient of the total host aid provided to the host individual's relatives. The first model mutualists in group s on individual host genotype in that group, i. Describes the association across space between formalizes this idea and introduces the notation and host genotype and total aid provided by hosts to mutual- approach for later extensions: ists. rmsi Regression coefficient of the total mutualist aid provided H = ,(1 + bi — ci ), (1) to hosts in group s on individual host genotype in that group, i. Describes the association across space between where Hsi is the abundance of the i-th host genotype host genotype and total aid provided by mutualists to hosts. in the s-th subgroup of the population, and the prime denotes abundance at one time step into the future. The role of demography The genotype i specifies a quantitative character that b, The fitness benefit to a mutualist individual per unit of aid determines the amount of aid given to mutualists. The available in the local group from hosts. c, The cost to hosts per unit aid given to the mutualist amount of aid, i, is determined by additive genes, and species. is is the average amount of aid given by hosts in the c2 The cost to mutualists per unit aid given to the host s-th group. The cost to the host per unit aid given to species. Vah The variance among groups in the average tendency of the mutualists is c, and the benefit that the mutualists hosts to give aid, the variance of is. return to the s-th group per unit aid received from the

ALTRUISM BETWEEN SPECIES 395

covariance. The average level of altruism increases when Cov(h,, i) > 0, which yields the condition known as Hamilton's rule (Grafen, 1985): risi risib — c > 0, (3) where ro is the regression of average group values, is, on individual values, i. This regression coefficient, ris,, is the standard coefficient of relatedness r from kin selection theory (Hamilton, 1970, 1972). This kin selection coefficient measures the genetic similarity between a donor, who gives up c units of reproduction, and the recipients in group s, who gain b units of reproduction. The FIG. 1. Diagram of the model given by eqn (1). coefficient ris, can also be viewed as a measure of genetic variability among groups. In the case here, i-th genotype is b. This is the standard linear additive where all genetic variability is additive, risi = ValV„ model of kin-selected altruism (Hamilton, 1970, 1972; where Va is the variance in genotypic values among Uyenoyama et al., 1981; Grafen, 1985; Wade, 1985; groups—the variance in i„ and V, is the total variance Queller, 1992). Notation for this and later sections is in genotypic values in the population—the variance in listed in Table 1. i over all groups. A picture of this interaction is given in Fig. 1. The level of aid to the mutualist, i, directly affects fitness, 2.2. GENETIC VARIABILITY IN BOTH HOST AND MUTUALIST h„, by the rate — c. Fitness is defined by eqn (1), with The previous model showed that spatial patterns of H s'i = H,h,. The genotypic value of an individual, i, genetic variability in the host determine the evolution affects the average genotypic value of the group, i„ by of mutualism. This section demonstrates that spatial the rate r. 1 ; this coefficient is defined below. A host correlations in the genetic variability of hosts and group with average genotype is provides aid to the mutualists can also influence the level of aid given to mutualists at a rate pi . The aid the mutualists provide mutualistic partners. The model is to hosts, A, is augmented at a rate p2 by aid received from the hosts; this extra benefit is given to all H H„(1 + bi is + b2js — ci), (4) members of the host subpopulation. The benefit to where j, is the average tendency of the s-th group the group per unit cost to an individual is bD=. 1.D 2, and the benefit to the original donor is rob. of the mutualists to provide benefit to the hosts. Behaviors are called altruistic if they cause individ- Using the Price equation as above, the condition for uals to give up some of their own direct reproduction an increase in the average level of aid given to in order to aid others (Hamilton, 1964). One way to mutualists, is study the evolution of altruistic behavior is to ask, for bi risi +b2 risi — c > 0, (5) given values of cost to an individual and benefit to neighbors per level of altruism, whether natural selec- where r s, is the regression of the mutualists' average tion favors an increase or decrease in the level of genotypic value for subgroups s on individual host altruism. We can use the expression in eqn (1) for the genotypes, i. fitness of a genotype with level of altruism i in an s The term b2 risi is the only change from the previous type subgroup, h„= 1 + bis — ci, to ask whether natu- model. This term arises from the correlation between ral selection causes the average level of altruism to host genotype i and the tendency of the local mutual- increase or decrease in the population. The simplest ists to give aid. The role of this correlation can be seen method for studying the evolution of altruism is the in Fig. 2. If hosts provide no direct benefit to mutu- Price equation (Price, 1970, 1972; Hamilton, 1970). alists, pi = 0, the benefit to their neighbors is The technique as described by Queller (1992) will be b1 = PiP2 = 0. The average level of altruism can, how- used here. ever, still increase by natural selection. This occurs According to the Price equation, when individuals with high values of i live in subgroups with mutualists that have a greater than fiAi = Cov(hs„ i ), (2) average genetic tendency to give aid, high j„ as where /i and T are the average values in the population measured by r si . In genetical language i is correlated for fitness and altruism, respectively, and Coy is the with a high fitness trait, which in this case happens to

396 S. A. FRANK

address this question a model is examined in which, initially, there is no correlation between species. Studying the course of selection within subgroups along with occasional migration episodes is tedious, so two limited goals are pursued. First, conditions are derived under which selection causes an increase in the average level of aid given from one species to its partner species. Second, conditions under which selection creates spatial associations between species, given initially uncorrelated distributions, are derived. The analysis could be extended to an explicit migration scheme, but selection-migration dynamics are not developed here. FIG. 2. Diagram of the model given by eqn (4). The aid provided The model of population dynamics and selection by the local group of mutualists, A, is affected by the average genotype of mutualists js at a rate of one. within each group s of the population is given by:

H = Hsi ( 1 + b2 Msis- ci i ) (8) be the trait of another species. The benefit gained per M s;' = Ms; ( 1 b, Hs is - c2j), (9) unit measure of risi is b2 = p2 (Fig. 2). where Ms; is the abundance of the j-th mutualist 2.3. THE ROLE OF ABUNDANCE genotype in the s -th group, and Hs and Ms are the The average tendency to give aid by donors, i s and total abundances of hosts and mutualists in the s -th js for hosts and mutualists, influenced the fitness group. In this model hosts are repaid for aid given to benefits of a host in the previous model. A more mutualists only after a time delay: the hosts aid the realistic assumption in many cases is that benefits mutualists, which increases the abundance of mutual- depend on the total value of aid available in the group ists, Ms , which then feeds back to provide a benefit rather than the average value per donor. Specifically, in the term b2 Msjs . A similar delay occurs from the the new model is mutualists' point of view. Thus a minimal model for the creation of spatial associations has two time H Hsi (1 + b1 Hs is + b2 Msjs - ci), (6) periods of selection in each group s, before processes where Hs and Ms are the host and mutualist abun- such as migration are allowed to mix the population dance in group s, so that Os = Hs is and its = Mjs are and destroy the developing associations. Specifically, total amounts of aid. The condition for an increase in the above equations must be expanded to give the the average aid given by hosts to mutualists, AT > 0, dynamics for Hsi and M s". . The analysis is given in the is Appendix. The condition for the increase in T, the average level b,rosi + b2 r,si - c > 0, (7) of aid given to mutualists by hosts, is (V' j where rosi is the regression of total host aid to risi b 2 [j b 2 )] mutualists in group s on individual host genotype in - 2c [1 - F jb ijb 2/2] > 0, (10) that group, and rms, is the regression of total mutualist aid to hosts in group s on individual host genotype in where VT is the variance among subpopulations in the that group. average tendency of mutualists to give aid, js . A number of simplifying assumptions were made to 2.4. THE ROLE OF DEMOGRAPHY obtain the condition (see Appendix). The most im- Spatial patterns of abundance and genetic variabil- portant assumptions are: host and mutualist abun- ity on both sides of a mutualistic interaction deter- dances and genotypes are spatially uncorrelated mine how mutualistic traits evolve. The previous before the first round of selection, so that risi = 0; models set the spatial patterns as parameters and did b, = b2 = b and c, = c2 = c; and c and risi are both not address how correlations within and between much smaller than b and j. Also, any negative density- species might arise. dependent effects occur globally rather than within Genetic correlations within species typically occur local subpopulations. Note that, because of the because relatives live near each other and share symmetry of the dynamical system in eqns (8) and (9), common ancestors. How might correlations in genetic conditions for change in j are easily obtained from variability and abundance arise between species? To eqn (10). ALTRUISM BETWEEN SPECIES 397

Increasing genetic variance of the mutualist among lated distributions (see Appendix). When the effects of subpopulations, , favors the spread of helping selection are weak, the correlation is approximately behavior in the host. Likewise, increasing genetic 2bcV ,„ c2VK, variance in the host among subpopulations, Va, fa- Corr(i ;,ja = ±bj_ co vors the spread of helping behavior in the mutualist. (1 + bl — Genetic variability among groups creates a spatial correlation between hosts and mutualists in the total where V%, is the average within group genetic variance, amount of aid per subpopulation. This may be one and for simplicity the parameters b, c and V,, are reason why genetic variability favors mutualism. The assumed to be equal for both species. association can be seen by the covariance after one The correlation of neutral loci between species will generation of selection between the total amount of remain zero after selection unless these neutral alleles aid by hosts, 4) s = H ;i ;, and the total amount of aid are linked with the altruism loci. by mutualists, = M :

Cov(4); 'Pa = ib l Va (1 — 2c,i) 3. Discussion

+ ib2 r: (1 — 2c2 j) + b, b2 \Tha r', (11) 3.1. SUMMARY OF MAIN RESULTS where the key assumptions are given below eqn (10), Natural selection favors helping another species only when benefits are returned to relatives of the and initial abundances are equal, H., = M„ before the first round of selection. (See the Appendix for as- original donor (Hamilton, 1972; Wilson, 1980). sumptions about the distribution of host and mutual- The first model formalizes this idea by an ist genotypes.) Some assumptions make it easier to see expression of Hamilton's rule for kin selection that what is required for a positive covariance. Let is appropriate for interspecific interactions [eqn (3)]. j = i = k and the values for b, c, and Va be the same In this kin selection model, benefits returned by for the two species, thus a mutualistic partner species are weighted by the coefficient of relatedness between the original donor Cov(4) , ) = 2bk-Va (1 — 2ck) + b 2V2a . and the recipients, risi . This coefficient is a type of genetic correlation between members of the same Small or moderate values of ck., cost multiplied by species. current average level of aid, yield a positive covari- The new models show that genetic associations ance. between species create an additional force favoring mutual aid between species [eqn (5)]. In this case a 2.5. BUILD-UP OF GENETIC CORRELATION BETWEEN strong tendency to give aid may be spatially corre- SPECIES lated with a strong tendency to receive aid. Reciproc- One key component of mutual aid between species ity is genetically determined, and natural selection is an association between the genotype of the host and favors an increase in mutual aid. the total value of aid by the local mutualists, rmsi [eqn The models also show that selection can create (7)]. The previous section showed that selection often positive correlations between species. For example, a causes a build-up in the association between total aid subpopulation of one species that is relatively gener- of hosts and mutualists, Cov(0 s , p > 0. Similar ous in giving aid to its partner species will have more analyses show that selection also tends to create partners from which to receive aid in the future. The positive associations between the genotype of one same is true from the partner's point of view. Thus species and the total value of aid given by the partner subpopulations with mutually generous individuals species, r us1 > 0 and r > 0. grow disproportionately large, creating a positive In this ecological model the total value of aid— correlation between species in tendency to give aid. abundance multiplied by average genotypic value— The only requirement of the model is that spatial determines the benefit returned for aid provided to a associations last long enough that aid given by an mutualistic partner. However, given the traditional individual to a partner species feeds back to provide interest in purely genetic correlations in studies of additional benefits to the donor's relatives. This may social behavior, a brief look at the build-up in genetic occur within or between generations. correlations between species is in order. The genetic correlation between species in average 3.2. INTERPRETATION genotypic value per subpopulation can build up after In what sense can aid given to another species be one generation of selection, given initially uncorre- considered "altruistic"? In the biological context, 398 S. A. FRANK

"altruism" is usually taken to mean sacrifice of an 3.3. EVOLUTIONARY DYNAMICS individual's own direct reproduction in order to aid Genetic correlation between species is determined the reproductive effort of another individual. It is by two opposing forces. Selection changes genetic customary to distinguish two types of altruism: kin variability within species into correlation between selection (Hamilton, 1964), in which aid is provided species, whereas migration of one species relative to to genetic relatives, and reciprocal altruism, in which its partner breaks down correlation. Thus a dynami- aid is provided to non-relatives of the same or cal analysis requires knowledge about how fast new different species (Trivers, 1971). According to this genetic variability within species is created relative to tradition, aid given to another species is reciprocal the rate at which migration destroys the correlation altruism if benefits are returned, or maladaptive altru- between species. ism if no benefits are returned. The dynamic tension between selection and mi- The altruism caused by the regression coefficients gration determines the maintenance of a spatial corre- between species, ris or rpsi , does not fit easily into a lation in mutual aid between species. It is possible, traditional category. On the one hand, these co- however, that spatial correlations played an import- efficients arise in the equations in exactly the same ant role only in the origin of mutual aid between way as the coefficients of relatedness within species: species, and that genetic variability does not influence both types of coefficient measure the fitness benefits the maintenance of these traits [see eqn (10)]. Specifi- caused by an association between the genotype of an cally, genetic variability and correlation between individual and the phenotypic properties of a group species may be required for the initial increase in the of neighbors. On the other hand, the coefficients average levels of aid provided by partner species. As between species have a relatively indirect effect: high these average levels increase, the condition for the altruism is favored when it is correlated with another maintenance or further increase of mutual aid be- high fitness trait, in this case the amount of altruism comes more easily satisfied. Correlations between given by neighbors of another species. Coefficients of species may no longer be needed to satisfy the con- relatedness within species are more direct: high altru- dition at this later stage. ism is favored when it confers benefits directly to The natural histories of symbiosis span the com- correlated genotypes. plete range of co-dispersal patterns, from vertically The coefficients between species measure reciproc- transmitted endosymbionts to widely dispersed ity. However, the reciprocity in this particular case is propagules of infectious diseases. The shared interests determined by a diffuse genetic and ecological corre- of hosts and their vertically transmitted endosym- lation rather than the complex behavioral mechan- bionts are well understood. The models presented isms between individuals typically associated with here show that this extreme form of co-dispersal is reciprocal altruism (Trivers, 1971). The uncertain just one end of a continuum. Natural selection builds status for the coefficients between species suggests correlations in genotype and abundance between either the need for a broader view, or a widening of more mobile partners. This creates varying degrees of old definitions and more elegant phrasing. (See shared reproductive interests between symbiotic Brown et al., 1982, for a definition of a coefficient of species across a wide range of natural communities. reciprocity in a Trivers-type model of intraspecific reciprocal altruism.) Both kin selection and associations between species I thank F. Breden and Y. Michalakis for helpful com- ments on the manuscript. My research is supported by create functional coherence for a group, where "func- National Science Foundation grant DEB-9057331 and tional coherence" means shared reproductive inter- National Institutes of Health grants GM42403 and BRSG- ests. This group-level coherence can be explained by S07-RR07008. stressing the fact that the selfish interests of the component individuals are aligned, or by stressing the REFERENCES fact that, as a unit, the group has beneficial traits that BARTON, N. H. & TURELLI, M. (1987). Adaptive landscapes, genetic the components lack when alone. There is a long a distance and the evolution of quantitative characters. Genet. Res. rather painful literature about this distinction 49, 157-173. (Brandon & Burian, 1984). The point here is that BRANDON, R. N. & BURIAN, R. M. (1984). Genes, Organisms, Populations: Controversies over the Units of Selection. Cam- selection can create associations between species and bridge, MA. MIT Press. the co-transmission of mutually favorable traits. This BROWN, J. S., SANDERSON, M. J. & MICHOD, R. E. (1982). Evolution causes either the functional coherence of the species of social behavior by reciprocation. J. theor. Biol. 99, 319-339. FRANK, S. A. (1986). Hierarchical selection theory and sex ratios I. pair, or the appearance of functional coherence, General solutions for structured populations. Theor. Popul. Biol. depending on one's tastes. 29, 312-342. ALTRUISM BETWEEN SPECIES 399

FRANK, S. A. (1987). Demography and sex ratio in social spiders. types in subpopulation s. The operator Es takes the Evolution 41, 1267-1281. GRAFEN, A. (1985). A geometric view of relatedness. Oxf. Surv. average over subpopulations s. evol. Biol. 2, 28-89. The first step is to obtain an expression for the total HAMILTON, W. D. (1964). The genetical evolution of social behav- change in each subpopulation after two time periods: ior. I, II. J. theor. Biol. 7, 1-52. HAMILTON, W. D. (1970). Selfish and spiteful behaviour in an evolutionary model. Nature, Lond. 228, 1218-1220. H's'i=H'„(1+b2Mj's—cli) HAMILTON, W. D. (1972). Altruism and related phenomena, mainly in social insects. A. Rev. ecol. Syst. 3, 193-232. =H„(1+b2Msjs—c,i) HAMILTON, W. D. (1975). Innate social aptitudes of man: an approach from evolutionary genetics. In: Biosocial Anthropology x (1 + b2 M s'j; —c,i)=H,h(sP, (Fox, R., ed.) pp. 133-155. New York: Wiley. KIMURA, M. (1965). A stochastic model concerning the mainten- where definitions for the terms can be found after eqn ance of genetic variability in quantitative characters. Proc. natn (8) in the main text. Acad. Sci. U.S.A. 54, 731-736. Next we need expressions for M 's and j; that are LANDE, R. (1975). The maintenance of genetic variability by mutation in a polygenic character with linked loci. Genet. Res. calculated from initial values in the subpopulation 26, 221-235. before the first round of selection: PRICE, G. R. (1970). Selection and covariance. Nature, Lond. 227, 520-521. M's = Ms (l+b,Hs is — c2js) PRICE, G. R. (1972). Extension of covariance selection mathematics. Ann. hum. Genet. 35, 485-490. QUELLER, D. C. (1992). A general model for kin selection. Evolution = Msm(s" 46, 376-380. SLATKIN, M. (1987). Heritable variation and heterozygosity under Ais a balance between mutations and stabilizing selection. Genet. =j5—c2Vsnlm(51), Res. 50, 53-62. TRIVERS, R. (1971). The evolution of reciprocal altruism. Q. Rev. Biol. 46, 35-57. where ms') is the total fitness of the mutualists in UYENOYAMA, M. K., FELDMAN, M. W. & MUELLER, L. D. (1981). subpopulation s after one round of selection, and VT Population genetic theory of kin selection: multiple alleles at one is the genetic variance of the mutualists in subpopu- locus. Proc. natn Acad. Sci. U.S.A. 78, 5036-5040. WADE, M. J. (1985). Soft selection, hard selection, kin selection, and lation s before the first round of selection. group selection. Am" 125, 61-73. Now we need an expression for the group fitness of WILLIAMS, G. C. (1960. Adaptation and Natural Selection. Prince- subpopulation s: ton, NJ: Princeton University Press. WILSON, D. S. (1980). The natural selection of populations and Hs = H;(1+ b2 Mj's — ciis) communities. Menlo Park, CA: Benjamin/Cummings. WILSON, D. S. (1983a). The effect of population structure on the =H5(1+b2Msjs—c,is)(1+b2M'ssis—c,i3 evolution of mutualism: a field test involving burying beetles and their phoretic mites. Am. Nat. 121, 851-870. =Hsh WILSON, D. S. (1983b). The group selection controversy: history and current status. A. Rev. ecol. Syst. 14, 159-189. i;= is + Ais WILSON, D. S. (1993). Complex interactions in metacommunities, with implications for biodiversity and higher levels of selection. =is — c,Vsh 117(2) Ecology 73, 1984-2000. h sn = (1 + b2Msjs—cii,),

APPENDIX where h 5(1) and h s(2) are the total fitness of the hosts in subpopulation s after one or two rounds of selection, Role of Demography respectively, and Vs is the genetic variance of the hosts The details are given here for the derivation of the in subpopulation s before the first round of selection. condition in eqn (10) in the main text. The model uses The condition for the increase in aid to the mutu- the recursions in eqns (8) and (9) in the text for two alist, AT> 0, is obtained by applying the expanded time periods of selection in each subpopulation. Price equation, eqn (A.1), and the following assump- The analysis here focuses on the host. Results for tions: the distributions of hosts and mutualists are the mutualist follow directly by the symmetry of the independent before selection; the initial abundance of model. To obtain information on selection both hosts and mutualists in each subpopulation before within and among subpopulations an expanded form selection is Hs = Ms = 1; the distribution of genotypic of the Price equation is used (Frank, 1986, 1987): values within subpopulations and among the averages of subpopulations are assumed to follow a normal EAT = Cov(h (s2) , is) + Es [Cov(h (.3), i )], (A.1) distribution so that the fourth central moment is where CP and h (s2) are the fitnesses after two time equal to three multiplied by the square of the vari- periods of selection for, respectively, host genotype i ance, the third central moment is zero, and subpopu- in subpopulation s, and the average over host geno- lation means are uncorrelated with subpopulation

400 S. A. FRANK variances. The assumption of a normal distribution is = —c2 V7, Cov(i,, 1/ms ) typical in quantitative genetic analyses, and the moment relations follow as properties of normality. See, — cl Vw Cov( j„ 1Ihs) for example, Kimura (1965), Lande (1975), Barton & + cl c2 Vw Vw Cov(1/hs , 1/m,), Turelli (1987), and Slatkin (1987). Following through, the condition for an increase in where Vni, and VmH, are the average within group I is variances for hosts and mutualists, respectively, and

r + b 3(b, — c1 )(V:i + j2 ) — 1 bj (1 + b 1 T) 125 =1 + b2j — ci f + b2 .54 — c16i, + 2'16 (1 + 2b2 j) - c;, (4V` + 312)] ms = 1 + bi — + bi —c264 - c, [2(1 - cl i + b2D+ b1 b2 ij — c2 b2 (r +P)]> 0, where 6is and (54 are independent normally distributed random variables with mean zero and variances where Va and V, are, respectively, the genetic variance ra and VT , respectively. The partitioned covariance among subpopulations, s, and the total variance in has three terms. The first term, Cov(is , Ajs ) = the population, and superscripts on V denote whether — c2 VmH, Cov(i„ llms ), is positive, because an increase the variance is for the mutualist, m, or the host, h. in is causes an increase in ms and thus a decrease in The main features of interest in this condition can lims . Similarly, the second term Cov( j„ Ais ) = be seen more easily when b1 = b2 = b, c, = c2 = c, and Cov( j„ 1Ihs ) is also positive. The third the order of magnitudes of c and r are approximately term, Cov(Ai„ Ajs )= c1 c2 r, VmH, Cov(1/h„ llms ), can the same, and much smaller than the magnitudes of be negative, because an increase in is or js within a b, T, and j. With these assumptions the condition group changes the fitnesses, h, and ms in opposite reduces to the expression given in eqn (10) in the main directions. Thus Cov(i s , j s ) will be positive when text. Cov(i„ Ajs) + Cov( j„ Ais )> Cov(Ai„ Ajs), and negative when the inequality is reversed. Build-up of Genetic Correlations I have not been able to derive an exact condition that determines whether the genetic covariance is This section shows how selection creates spatial positive or negative. If we assume that the effects of correlations in genotypic value between species. the interactions are weak, that is, hs ',:dd 1 and ms ',:sd 1 Changes by selection are studied given that initial and the genetic variances are small because (5, «1 and genotypic distributions of host and mutualists are 454 « 1, then an approximation for the genetic covari- independent, Cov(is ,js )= 0. The selection equations ance can be obtained by taking the Taylor series are eqns (8) and (9) in the main text. Assumptions expansions of 11h, and 1/m, in bis and .54 and keeping about the distributional properties of genotypes be- only the linear and second-order terms, yielding fore selection are given in the previous section. The condition of interest is Cov(i a> 0. The . . , 2bcV„,V,, 1 c2V„, Cov (r„ s ) first step is to break the covariance into manageable (1 + b.1 — ci)2 (1 + bj — i) pieces: where, for simplicity, the parameters b, c, VH, and Va Cov(i ,.i = Cov(is + Ais, is + A.h) are assumed to be equal for both species. The variance among groups after selection, V a' , will typically = Cov(is , Afs ) + Cov(is, be smaller than before selection, Va , therefore the + Cov(Ai„ Ajs) correlation between species is at least Cov(i 's ,j;)/V,, .